#8094832

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(4,8)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,2,5,3)(4,6,8,7)(9,11,15,12)(10,16,19,17)(13,20,18,22)(14,21,24,23)(25,27,30,26)(28,31,36,33)(29,32,37,34)(35,39,40,38);(2,3)(6,7)(9,11,15,12)(10,17,19,16)(13,22,18,20)(14,21,24,23)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,12,15,11)(10,17,19,16)(13,22,18,20)(14,23,24,21)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40);(1,3,5,2)(4,7,8,6)(9,13,15,18)(10,14,19,24)(11,20,12,22)(16,21,17,23);(1,4)(2,6)(3,7)(5,8)(9,14)(10,18)(11,21)(12,23)(13,19)(15,24)(16,22)(17,20)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002028);